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The Garden of Knowledge as a Knowledge Manifold  A Conceptual Framework for Computer Supported Subjective Education
 CID17, TRITANAD9708, DEPARTMENT OF NUMERICAL ANALYSIS AND COMPUTING SCIENCE
, 1997
"... This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in te ..."
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This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols  procedures that are implemented on top of the participating subjective knowledgepatches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inversedirect limit duality of Category Theory.
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Mathematical masterpieces: teaching with original sources
 Vita Mathematica: Historical Research and Integrations with Teaching
, 1996
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The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
Cantor's Grundlagen and the Paradoxes of Set Theory
"... This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 197 ..."
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This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motivated my first attempts to understand proper classes and the realm of transfinite numbers. I read a version of the paper at the APA Central Division meeting in Chicago in May, 1998. I thank Howard Stein, who provided valuable criticisms of an earlier draft, ranging from the correction of spelling mistakes, through important historical remarks, to the correction of a mathematical mistake, and Patricia Blanchette, who commented on the paper at the APA meeting and raised two challenging points which have led to improvements in this final version
Notes on Discrete Mathematics for Computer Scientists
, 2003
"... 1.2 Formal Languages.......................... 2 ..."
Completions of ordered algebraic structures: a survey
 Proceedings of the International Workshop on Interval/Probabilistic Uncertainty and Nonclassical Logics. Advances in Soft Computing
"... Summary. Ordered algebraic structures are encountered in many areas of mathematics. One frequently wishes to embed a given ordered algebraic structure into a complete ordered algebraic structure in a manner that preserves some aspects of the algebraic and order theoretic properties of the original. ..."
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Summary. Ordered algebraic structures are encountered in many areas of mathematics. One frequently wishes to embed a given ordered algebraic structure into a complete ordered algebraic structure in a manner that preserves some aspects of the algebraic and order theoretic properties of the original. It is the purpose here to survey some recent results in this area. 1
On the education of mathematics majors
 Contemp. Issues Math. Ed
, 1999
"... Upper division courses in college are where math majors learn real mathematics. For the first time they get to examine the foundations of algebra, geometry and analysis, come facetoface with the deductive nature of mathematics on a consistent basis and, most importantly, learn to do serious theore ..."
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Upper division courses in college are where math majors learn real mathematics. For the first time they get to examine the foundations of algebra, geometry and analysis, come facetoface with the deductive nature of mathematics on a consistent basis and, most importantly, learn to do serious theoremproving. For reasons not unlike these, most mathematicians enjoy teaching these courses more than others. While teaching graduate courses may be professionally more satisfying, it also involves more work, and the teaching of lower division courses — calculus and elementary discrete mathematics — is a strenuous exercise in the suppression of one’s basic mathematical impulses. By contrast, the teaching of upper division courses involves no more than doing elementary mathematics the usual way: abstract definitions can be offered without apology and theorems are proved as a matter of principle. This is something we can all do on automatic pilot. But have we been on automatic pilot for too long? Mathematicians approach these courses as a training ground for future mathematicians. Even a casual perusal of the existing textbooks would readily confirm this fact. We look at upper division courses as the first steps of a journey of ten thousand miles: in order to give students a firm foundation for future research, we feed them technicality after technicality. If they do not fully grasp some of the things they are taught, they will when they get to graduate school or, if necessary, a few years after they start their research. Then they will put everything together. In short, we build the undergraduate education program for our majors on the principle of delayed gratification. Whatever their misgivings for the time being, students will benefit in the long haul. This is the abbreviated version of a longer paper [W4] which discusses the same issues from a slightly different perspective, that of training prospective school teachers. I am indebted to